2. Formulae for dust-driven winds
The optical depth is proportional to the column density and dust-to-gas mass ratio . The column density is proportional to the total mass density and inner radius of the dusty CSE. The dust grains in the dust-formation zone are assumed to be in radiative equilibrium with the incident stellar radiation field, keeping the effective temperature and the dust condensation temperature fixed. Then , the stellar luminosity. The continuity equation yields , with gas+dust mass-loss rate , and expansion velocity . Here the drift velocity - the velocity difference between the gas and dust fluids - is neglected, i.e. gas and dust are assumed to be well coupled (cf. Lamers & Cassinelli 1999). Gas pressure and wind-driving mechanisms other than radiation pressure on dust are neglected as well, although they become important for wind speeds below a few km s-1 (e.g. Steffen et al. 1997, 1998). It follows that the optical depth
The constant of proportionality includes a factor , the wavelength-dependent opacity of the dust, and a temperature dependence as .
In a radiation-driven outflow the matter-momentum flux is coupled with the stellar photon-momentum flux via the momentum equation (Gail & Sedlmayr 1986)
with the flux-weighted optical depth. In general, the ratio of and depends on the mass-loss rate. Detailed computations such as those presented in Habing et al. (1994), however, show that this dependence vanishes for yr-1, which applies to the class of obscured AGB stars under study here (van Loon et al. 1999b). Combination of Eqs. (1) and (2) then leads to a description of the expansion velocity in terms of dust-to-gas ratio and luminosity:
Eq. (2) can also be used to eliminate from Eq. (1), yielding a relation between the mass-loss rate and dust-to-gas ratio, and the optical depth and luminosity:
Alternatively, Eq. (2) can be used to eliminate L from Eq. (1), yielding a relation between the mass-loss rate and dust-to-gas ratio, and the optical depth and expansion velocity:
For magellanic stars luminosities are easier to measure than expansion velocities, and for these stars it is advantageous to make use of Eq. (4). For galactic stars the opposite is true, and for them Eq. (5) is the formula to use.
The constants in Eqs. (4) and (5) are related. Calling the constants of proportionality of Eqs. (1) and (2) respectively and , the constant in Eq. (4) equals and the constant in Eq. (5) equals . The constant of proportionality in Eq. (3) equals . The values of and depend on the properties of the dust species, and calibrating them is an important yet extremely difficult task that will not be exercised here.
© European Southern Observatory (ESO) 2000
Online publication: January 31, 2000